Finite Group Databasebeta

← #36714673

Subgroups — #36714673

Order 22784.

2314 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#13083575C211yesmaximal, characteristic
2#26023227C213yesmaximal, complemented
3#26023223C213yesmaximal, complemented
4#26023173C211yesmaximal, characteristic, complemented
5#26023136C216yesmaximal, complemented
6#26023090C212yesmaximal
7#26022891C216yesmaximal, complemented
8#26022648C213yesmaximal, complemented
9#26022645C213yesmaximal, complemented
10#26022617C212yesmaximal, complemented
11C89×#2351C211yesmaximal, characteristic, complemented
12C22×#13083575C2211yescharacteristic
13C4×#13083575C4116yescomplemented
14C2×Q8×Dic89C2212yes—
15C2×Q8×Dic89C2211yescharacteristic
16C2×C4×D356C2213yes—
17C2×#15667974C2213yes—
18C2×#15667970C2213yes—
19C2×C42×D89C2213yes—
20#19151500C2216yes—
21#19151499C2216yes—
22#19151497C2216yes—
23#19151490C22112yes—
24#19151488C22112yes—
25#19151465C2213yes—
26#19151465C2211yescharacteristic
27#19151464C2213yes—
28#19151463C2216yes—
29#19151460C2213yes—
30#19151460C2213yes—
31#19151460C2216yes—
32#19151455C2216yes—
33#19151455C2216yes—
34C4⋊C4×Dic89C2216yes—
35C4⋊C4×Dic89C2216yes—
36C4×C4⋊Dic89C2213yes—
37C4×C4⋊Dic89C2216yes—
38C4×#13083551C2216yes—
39#19151445C2216yes—
40#19151444C2216yes—
41#19151443C2216yes—
42C2×C4×Q8×C89C2212yes—
43C2×C4×Q8×C89C2211yescharacteristic
44#389×C89C2216yes—
45#382×C89C2213yes—
46C4×C4⋊C4×C89C2213yes—
47C2×#13083575—24no—
48C2×#13083575—24no—
49C2×#13083575C2×C418yes—
50C22×D356C2313yes—
51C22×C4×D89C2313yes—
52C22×C4×D89C2313yes—
53C22×C4×D89C2313yes—
54Q8×Dic89—28no—
55Q8×Dic89C2×C418yes—
56C4×D356C2×C4124yes—
57#15667974C2×C4124yes—
58#15667970C2×C4124yes—
59C2×C4⋊Dic89C2316yes—
60C2×C4⋊Dic89C2316yes—
61C2×C4⋊Dic89C2313yes—
62C2×#13083551C23112yes—
63C2×#13083551C2316yes—
64C2×#13083551C2316yes—
65C2×#13083551C2316yes—
66C2×C4×Dic89C2316yes—
67C2×C4×Dic89C23112yes—
68C2×C4×Dic89C2313yes—
69C2×C4×Dic89C2313yes—
70C2×C4×Dic89C2311yescharacteristic
71C42×D89C2×C4124yes—
72#15667919C2316yes—
73#15667919C2316yes—
74#15667919C2312yes—
75#15667919C2316yes—
76#15667919C2316yes—
77#15667919C2316yes—
78#15667919C2313yes—
79#15667919C2312yes—
80#15667919C2311yescharacteristic
81C22×Q8×C89C2311yescharacteristic
82C4×Q8×C89—28no—
83C4×Q8×C89C2×C418yes—
84C2×C4⋊C4×C89C2316yes—
85C2×C4⋊C4×C89C2316yes—
86C2×C4⋊C4×C89C2313yes—
87C2×C42×C89C2316yes—
88C2×C42×C89C2313yes—
89#95×C89C2311yescharacteristic
90#95×C89C2313yes—
91#95×C89C2316yes—
92C23×D89C2413yes—
93#13083575—232no—
94C2×D356—26no—
95C2×D356—26no—
96C2×D356C22×C4112yes—
97C22×Dic89C2412yes—
98C22×Dic89C2416yes—
99C22×Dic89C2411yescharacteristic
100C22×Dic89C2411yescharacteristic
101C22×Dic89C2413yes—
102C2×C4×D89—26no—
103C2×C4×D89—26no—
104C2×C4×D89—46no—
105C2×C4×D89—46no—
106C2×C4×D89C22×C4112yes—
107C2×C4×D89C22×C4112yes—
108C2×C4×D89C22×C4112yes—
109C4⋊Dic89—212no—
110C4⋊Dic89—46no—
111C4⋊Dic89C22×C4112yes—
112#13083551—46no—
113#13083551—46no—
114#13083551—412no—
115#13083551C22×C4124yes—
116C4×Dic89—212no—
117C4×Dic89—412no—
118C4×Dic89C22×C4112yes—
119C4×Dic89C22×C4112yes—
120C4×Dic89C22×C414yes—
121C2×Q8×C89C22×C414yes—
122C2×Q8×C89#4114yes—
123C2×Q8×C89#4114yes—
124C22×C4×C89C2416yes—
125C22×C4×C89C2412yes—
126C22×C4×C89C2411yescharacteristic
127C22×C4×C89C2413yes—
128C22×C4×C89C2413yes—
129C4⋊C4×C89—212no—
130C4⋊C4×C89—46no—
131C4⋊C4×C89C22×C4112yes—
132C42×C89—212no—
133C42×C89C22×C4112yes—
134C23×C89C2511yescharacteristic
135C22×D89—23no—
136C22×D89—23no—
137C22×D89—26no—
138C22×D89—26no—
139C22×D89C23×C416yes—
140Q8×C89#117116yes—
141C2×C4×C89—26no—
142C2×C4×C89—22no—
143C2×C4×C89—22no—
144C2×C4×C89—26no—
145C2×C4×C89—22no—
146C2×C4×C89—26no—
147C2×C4×C89—26no—
148C2×C4×C89—22no—
149C2×C4×C89C23×C412yes—
150C2×C4×C89C23×C416yes—
151C2×C4×C89C23×C416yes—
152C2×C4×C89C2×#4116yes—
153C2×C4×C89C2×#4116yes—
154C2×Dic89—21no—
155C2×Dic89—21no—
156C2×Dic89—26no—
157C2×Dic89—22no—
158C2×Dic89—22no—
159C2×Dic89—26no—
160C2×Dic89—26no—
161C2×Dic89—22no—
162C2×Dic89—26no—
163C2×Dic89—22no—
164C2×Dic89C23×C412yes—
165C2×Dic89C23×C412yes—
166C2×Dic89C23×C416yes—
167D356—224no—
168C4×D89—224no—
169C4×D89—412no—
170C4×D89—412no—
171C22×C89C24×C411yescharacteristic
172C22×C89C22×#4111yescharacteristic
173C22×C89C22×#4111yescharacteristic
174C22×C89#58512yes—
175C22×C89#58512yes—
176C2×D89—26no—
177C2×D89—26no—
178C2×D89—26no—
179C2×D89—46no—
180C2×D89—46no—
181C4×C89—22no—
182C4×C89—26no—
183C4×C89—42no—
184C4×C89—46no—
185C4×C89C2×#117112yes—
186Dic89—22no—
187Dic89—22no—
188Dic89—26no—
189Dic89—42no—
190Dic89—46no—
191#34197—891nomaximal, Sylow
192C2×C89C22×#11711yescharacteristic
193C2×C89#342811yescharacteristic
194C2×C89#342811yescharacteristic
195C2×C89#352412yes—
196C2×C89#352412yes—
197D89—46no—
198C2×C4×#41—891no—
199#2357—893no—
200#2357—893no—
201#2351—891no—
202#2351D8911yescharacteristic, complemented
203#2346—896no—
204#2341—892no—
205#2319—896no—
206#2283—893no—
207#2283—893no—
208#2276—892no—
209C89#3419711yescharacteristic, complemented, Sylow
210C22×#41—891no—
211C4×#41—8916no—
212C2×C4×Q8—891no—
213C2×C4×Q8—892no—
214C2×C4×Q8C2×D8912yes—
215C2×C4×Q8C2×D8911yescharacteristic
216C2×C4×D4—893no—
217C2×C4×D4—893no—
218C2×#117—893no—
219C22×C42—893no—
220#390—896no—
221#390—896no—
222#389—896no—
223#389C2×D8916yes—
224#388—8912no—
225#388—8912no—
226#383—891no—
227#383—893no—
228#382—896no—
229#382—893no—
230#382C2×D8913yes—
231#381—896no—
232#381—893no—
233#381—893no—
234#380—896no—
235#380—896no—
236C4×C4⋊C4—896no—
237C4×C4⋊C4—896no—
238C4×C4⋊C4—896no—
239C4×C4⋊C4—893no—
240C4×C4⋊C4C2×D8913yes—
241C4×C22⋊C4—896no—
242C4×C22⋊C4—896no—
243#376—896no—
244#376—896no—
245C2×#41—898no—
246C2×#41—1784no—
247C2×#41—1784no—
248C22×Q8C22×D8911yescharacteristic
249C22×D4—893no—
250C23×C4—893no—
251C23×C4—893no—
252C23×C4—893no—
253C4×Q8—28no—
254C4×Q8—898no—
255C4×Q8—1788no—
256C4×Q8C4×D8918yes—
257C4×D4—8924no—
258C4×D4—8924no—
259#117—8924no—
260C2×C4⋊C4—893no—
261C2×C4⋊C4—896no—
262C2×C4⋊C4—896no—
263C2×C4⋊C4C22×D8916yes—
264C2×C4⋊C4C22×D8916yes—
265C2×C4⋊C4C22×D8913yes—
266C2×C22⋊C4—896no—
267C2×C22⋊C4—896no—
268C2×C22⋊C4—896no—
269C2×C22⋊C4—8912no—
270C2×C42—893no—
271C2×C42—893no—
272C2×C42—891no—
273C2×C42—8924no—
274C2×C42—8912no—
275C2×C42—896no—
276C2×C42C22×D8916yes—
277C2×C42C22×D8913yes—
278#95—892no—
279#95—896no—
280#95—896no—
281#95—896no—
282#95—896no—
283#95—896no—
284#95—892no—
285#95—893no—
286#95—891no—
287#95C22×D8916yes—
288#95C22×D8913yes—
289#95C22×D8911yescharacteristic
290C24—893no—
291#41—17832no—
292C2×Q8C2×C4×D8914yes—
293C2×Q8#1308357314yes—
294C2×Q8#1308357514yes—
295C2×D4—8912no—
296C2×D4—1786no—
297C2×D4—1786no—
298C22×C4—891no—
299C22×C4—891no—
300C22×C4—893no—
301C22×C4—8912no—
302C22×C4—8912no—
303C22×C4—8912no—
304C22×C4—892no—
305C22×C4—896no—
306C22×C4—3566no—
307C22×C4—3566no—
308C22×C4—1786no—
309C22×C4—1786no—
310C22×C4C23×D8912yes—
311C22×C4C23×D8916yes—
312C22×C4C23×D8911yescharacteristic
313C22×C4C23×D8913yes—
314C22×C4C23×D8913yes—
315C4⋊C4—212no—
316C4⋊C4—46no—
317C4⋊C4—8912no—
318C4⋊C4—3566no—
319C4⋊C4—17812no—
320C4⋊C4C2×C4×D89112yes—
321C22⋊C4—8924no—
322C22⋊C4—35612no—
323C22⋊C4—3566no—
324C22⋊C4—3566no—
325C42—212no—
326C42—894no—
327C42—8912no—
328C42—8912no—
329C42—35612no—
330C42—17812no—
331C42C2×C4×D89112yes—
332C23—896no—
333C23—1783no—
334C23—1783no—
335C23—1786no—
336C23—1786no—
337C23C24×D8911yescharacteristic
338Q8#15667970116yes—
339D4—17824no—
340C2×C4—26no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—22no—
346C2×C4—26no—
347C2×C4—22no—
348C2×C4—892no—
349C2×C4—892no—
350C2×C4—896no—
351C2×C4—35612no—
352C2×C4—35612no—
353C2×C4—1782no—
354C2×C4—1786no—
355C2×C4—1786no—
356C2×C4—1782no—
357C2×C4—1786no—
358C2×C4—1782no—
359C2×C4—1781no—
360C2×C4—1781no—
361C2×C4—1782no—
362C2×C4—1786no—
363C2×C4—17824no—
364C2×C4C22×C4×D8912yes—
365C2×C4C22×C4×D8916yes—
366C2×C4C22×C4×D8916yes—
367C2×C4C2×#1308357316yes—
368C2×C4C2×#1308357516yes—
369C22—3566no—
370C22—3566no—
371C22—1786no—
372C22—1786no—
373C22—1786no—
374C22C23×C4×D8911yescharacteristic
375C22C22×#1308357311yescharacteristic
376C22C22×#1308357511yescharacteristic
377C22#1915243412yes—
378C22#1915243512yes—
379C4—26no—
380C4—22no—
381C4—42no—
382C4—46no—
383C4—3562no—
384C4—3566no—
385C4—1782no—
386C4—1782no—
387C4—1786no—
388C4C2×#15667970112yes—
389C2—3566no—
390C2C22×#1566797011yescharacteristic
391C2#2603345111yescharacteristic
392C2#2603345411yescharacteristic
393C2#2603435412yes—
394C2#2603436412yes—
subgroupG/H
center, FrattiniC23C24×D89
commutatorC22×C89C24×C4
FittingC89×#2351C2
socle, cosocleC23×C89C25
prime radicalC22×#13083575C22
hypercenter#2351D89
hypocenterC89#34197
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ #2351

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26023227C2semidirect
3. #26023223C2semidirect
4. #26023173C2semidirect
5. #26023136C2semidirect
7. #26022891C2semidirect
8. #26022648C2semidirect
9. #26022645C2semidirect
10. #26022617C2semidirect
11. C89×#2351C2semidirect
13. C4×#13083575C4semidirect
202. #2351D89semidirect
209. C89#34197semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×#130835752
#260232272
#260232232
#260231732
#260231362
#260230902
#260228912
#260226482
#260226452
#260226172
C89×#23512
#3419789